Why there is gap between 2 face
The reason there is a gap is because when you use a modifier, the mesh editor is not aware of the other faces, and therefore they are not joined. As a result the face will behave like it is not connected. To get the behavior you desire, you will need to extrude by scaling.


Select the vertex at the center of your mesh
hit Shift-S>Cursor to selected
Set the pivot point to 3D cursor. (.)
Hit E to extrude, then S to scale, and Shift-Z to stop it from expanding vertically. Scale to taste.


Gif:



Note: There will be interior faces on the sides of your new extrusion, it would probably be best to remove them.